Nodules and concretions in general often form around a nucleus, which can be anything different from the surrounding rock - such as a fossil. In your specimen I can't tell what the fossil is.

In general:

1. Sandstone (and other sedimentary rocks) are composed of particles and cement. The cement arises post-deposition, by crystallization from fluids in the pores, and it is that which holds the rock together (though, there are other mechanisms that do the same thing).

2. Cementation is one of the processes that happens in the early stages of lithification. In this phase (diagensis), the physical conditions are changing from those pertaining when the sediment was deposited. As a result there are tendencies for the minerals making up the particles and/or cement to change their chemical and physical structure (striving to maintain equilibrium). Such changes may be evenly distributed through the evolving rock, or they may be exaccerbated here and there due to local anomalies. The latter can manifest itself as concretions.

3. Concretions are a little like rain drops, in that they may form spontaneously in clean air, or they may nucleate on an anomaly, such as dust or pollen in the case of rain. With concretions, it may be a fossil or an abnormal mineral grain that serves to get it started.

4. As the concretion grows, chemical changes are likely to be continuing, and not necessarily in a smooth way. That can account for the concentric banding in (some) concretions. Here's an example

5. Depending on the history of the sediment-rock, the concretion may eventually stop growing, becoming frozen in time, possibly to be found millions of years later when the rock is exposed after all sorts of geological events. If it is sufficiently disctinct from its host rock, it may become loose and separate.

6. Now it is subject to rather different physical conditions to when it was growing and while it was lithified. So further chemical and physical changes occur - looseley referred to as weathering.